Computing connectedness: Disconnectedness and discreteness

We consider finite point-set approximations of a manifold or fractal with the goal of determining topological properties of the underlying set. We use the minimal spanning tree of the finite set of points to compute the number and size of its ∈-connected components. By extrapolating the limiting behavior of these quantities as ∈ → 0 we can say whether the underlying set appears to be connected, totally disconnected, or perfect. We demonstrate the effectiveness of our techniques for a number of examples, including a family of fractals related to the Sierpinski triangle, Cantor subsets of the plane, the Hénon attractor, and cantori from four-dimensional symplectic sawtooth maps. For zero-measure Cantor sets, we conjecture that the growth rate of the number of ∈-components as ∈ → 0 is equivalent to the box-counting dimension.